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Mode{{ resultsReady ? coordinateModeLabel : '—' }} Samples{{ resultsReady ? formatInteger(computation.values.sample_count) : '—' }} Visible points{{ resultsReady ? formatInteger(computation.values.plot.visible_point_count) : '—' }}
Polar angles are measured in radians.
Enter up to four functions, one per line. Use {{ variableName }}; the {{ equationPrefix }} prefix is optional.
{{ issueFor('expressions') }}
The sampled domain must start below its maximum.
Domain fields accept decimal numbers; constants such as pi belong in expressions.
Samples outside the visible window remain available in exports.
The visible window must end above its minimum.
The neutral default is 601 evenly spaced samples per function.

The chart renderer is unavailable. The audit and sample tables remain available.

FunctionFiniteInvalidMinimumMaximumCopy
{{ row.source }}{{ formatInteger(row.finite) }}{{ formatInteger(row.invalid) }}{{ formatNumber(row.minimum) }}{{ formatNumber(row.maximum) }}
Function{{ variableName }}OutputPlot xPlot yWindowCopy
{{ row.source }}{{ formatNumber(row.input) }}{{ formatNumber(row.output) }}{{ formatNumber(row.x) }}{{ formatNumber(row.y) }}{{ row.visible ? 'Visible' : 'Outside' }}
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A function plot turns an input-output relationship into a shape. In Cartesian coordinates, each input x produces a vertical value y. In polar coordinates, an angle produces a radius, and that radius is converted to a point around the origin. The same expression can therefore describe very different geometry depending on the coordinate system.

Graphs help reveal intercepts, turning points, growth, oscillation, symmetry, and regions where a function is undefined. They are also approximations whenever software evaluates only a finite set of inputs. A narrow peak can fall between samples, a vertical asymptote can look like a connecting line, and a visible window can hide valid values without changing the function itself.

Domain
The input interval that is evaluated, including both endpoints.
Visible window
The vertical Cartesian range, or the square x-y range used to display a polar curve.
Sample
One evaluated input and its finite output, converted to a plotted point when possible.
Discontinuity
A break, jump, asymptote, or undefined location where a continuous line would be misleading.

Sampling density should match the behavior being investigated. Smooth lines and broad parabolas may need few points, while high-frequency trigonometric curves or narrow features need more. More samples improve resolution but do not perform symbolic analysis or prove that every discontinuity has been found.

A graph is strongest when the expression, domain, and point table agree. Use the plot to see shape, then check finite and invalid counts and individual sampled values before drawing a conclusion from a crossing, gap, or apparent extremum.

How to Use This Tool:

Choose the coordinate system first because it determines both the variable and the meaning of the domain.

  1. Select Cartesian — y = f(x) or Polar — r = f(theta). Polar angles are measured in radians.
  2. Enter one to four functions, one per line. Use x in Cartesian mode or theta or t in polar mode; an equation prefix such as y = or r = is optional.
  3. Set the domain minimum and maximum, then choose the visible minimum and maximum. Each minimum must be less than its matching maximum.
  4. Open Advanced to change Samples per function from 3 to 1,001. The default is 601 evenly spaced samples.
  5. Correct any expression or range error, then inspect Function plot, Series audit, and the sampled rows that matter to the feature you are studying.

Interpreting Results:

The plot shows only finite points that fall within the selected window. A point outside the window remains in the sampled data, while an undefined or infinite result is counted as invalid and creates no plotted coordinate.

  • Use Finite and Invalid counts to distinguish a clipped curve from a function that cannot be evaluated at some inputs.
  • Use the minimum and maximum sampled outputs as evidence about the chosen grid, not guaranteed global extrema between samples.
  • Check the Sample ledger around a suspicious jump or crossing. A visual gap can be real, while a connecting segment can still hide an unsampled asymptote.
  • Compare runs only when the expression, coordinate mode, domain, visible window, and sample count stay the same.

Technical Details:

Expressions are parsed into a bounded arithmetic tree before evaluation. Supported syntax includes decimal and scientific notation, parentheses, implicit multiplication, the constants pi and e, arithmetic operators, and a defined set of trigonometric, logarithmic, rounding, power, minimum, maximum, and distance functions. Assignment chains, unknown names, and unsupported characters are rejected.

Formula Core:

With N samples over the closed domain from a to b, adjacent inputs are evenly spaced. The final input is set exactly to b to avoid accumulated step error.

Δu=b-aN-1 ui=a+iΔu,i=0,…,N-1 x=r×cos(θ),y=r×sin(θ)

For Cartesian mode, u is x and the plotted point is (x, f(x)). For polar mode, u is the radian angle θ, r is the expression output, and the last equation converts the sample to Cartesian display coordinates.

Rule Core:

RuleExact behaviorConsequence
Expression limitUp to four non-empty lines, 500 characters total, and 120 characters per line.Longer or extra expressions are rejected.
Numeric boundsDomain and visible-window values must be finite decimals from −1,000,000 to 1,000,000.Constants such as pi belong in expressions, not range fields.
Finite-point ruleNaN and infinite outputs do not become coordinates.A run fails if every sampled value is non-finite.
Visibility ruleCartesian points require y inside the visible range; polar points require both converted x and y inside it.A run fails when no sampled point is visible.
Likely break ruleIn Cartesian mode, a segment is broken when consecutive large, opposite-sign outputs jump by more than 1.5 times the visible span while both lie far from its center.This reduces some false lines across asymptotes but cannot detect every discontinuity.

Calculations use double-precision floating-point numbers. Trigonometric functions use radians, and some library results can differ slightly across browsers, operating systems, or processor architectures.

Accuracy Notes:

Finite sampling measures the function only at the chosen inputs. It does not solve equations, differentiate, integrate, find exact extrema, draw implicit equations, or prove continuity.

  • Increase the sample count or narrow the domain when a feature may be smaller than the current spacing.
  • Change the visible window when finite values exist but the curve is clipped.
  • Confirm consequential roots, extrema, and asymptotes with analytic work or a dedicated numerical method.

Worked Examples:

Three-point line check

For 2*x + 1 over −1 to 1 with three samples, the spacing is 1. The points are (−1, −1), (0, 1), and (1, 3), so the audit reports three finite points and no invalid values.

Polar half-circle

For radius 1 from 0 to pi with three samples, the angles are 0, π/2, and π radians. They convert approximately to (1, 0), (0, 1), and (−1, 0), tracing the upper half of a unit circle.

References: